Part 1 - AL-Tax

Chapter 3W is an NT NT block-diagonal spatial-weighting matrix (with elements w ij ).Thus, Wy is the spatial lag, given in scalar notation as the first term on the righthandside of equation (3.2). The diagonal elements of the off-diagonal T Tblocks in W, which reflect the contemporaneous effect of the column unit on therow unit, are the w ij that reflect the degree of connection from unit j to i – so,unlike a variance–covariance matrix, W need not be symmetric. ρ, the spatialautoregressive coefficient, reflects the impact of the outcomes in the other (j i)spatial units, as weighted by w ij , on the outcome in i. Thus, ρ gauges the overallstrength of diffusion, whereas the w ij describe the relative magnitudes of thediffusion paths between the sample units.Generally, the set of w ij is determined by theoretical and substantive argumentationas to which units will have greatest affect on outcomes in other units;ρ values are the coefficients to be estimated on these spatial lags. For example,operationalization of the tax-competition argument would be weights, w ij , basedon the trade or capital-flow shares of countries j in country i’s total. The innerproduct of that vector of weights with the stacked dependent variable y thengives the weighted sum (or average) of y in the other countries j in that timeperiodas a right-hand-side variable in the regression. The matrix Wy just givesthe entire set of these vector inner products – in this case, the trade- or capitalflow-weightedaverages – for all countries i. 7 X is a matrix of NT observations onK exogenous regressors – in our case, η, ξ, and η ξ– β is a K 1 vector of coefficientsthereupon, and ε is an NT 1 vector of residuals, with the usual propertiesassumed.In Franzese and Hays (2004, 2006), we demonstrate analytically, in the simplestpossible case (one domestic factor, X, two countries, 1 and 2, and conditionallyi.i.d. errors, ε) that OLS estimates of equation (3.7) (or the identical (3.8))will suffer simultaneity bias and, obviously, that OLS estimates of equation (3.7)omitting the spatial lag will suffer omitted-variable bias, and we specify thosebiases insofar as possible.This simple case highlights that OLS estimates of equation (3.7) will suffersimultaneity (endogeneity) bias:Y β X ρY ε1 1 112 2 1Y β X ρ Y ε .2 2 2 21 1 2(3.9)(3.10)The left-hand side of equation (3.9) is on the right-hand side of equation (3.10)and vice versa: Textbook endogeneity. In words: Country 2 affects country 1, but55